Amer. Funct. However checking the induction process carefully, it is possible to give the constants explicitly. A Laurent polynomial is a Laurent series for which all but finitely many f nf_n are zero. It would perhaps be clearer if we used the term “restricted Laurent series” to cover the Laurent series considered in Definition , and let “Laurent series” be the term that covers doubly infinite series. The ring of formal Laurent series over a commutative ring AA in an indeterminate xx consists of Laurent series ∑ n∈ℤf nz n\sum_{n \in \mathbb{Z}} f_n z^n, with f n∈Af_n \in A but where all but finitely many f nf_n for n<0n \lt 0 vanish. Sum of natural numbers of zeta function Z(-1)=1+2+3+ ... Then we express the function as the Laurent series at q=0 by using the function A(t) and C(t). on Number Theory, pp. \({\mathbf {m}}\in ({\mathbb {Z}}_{\le 0})^r\), \(\varepsilon _1=s_1-m_1,\ldots ,\varepsilon _r=s_r-m_r\), \(\varepsilon =\max \{|\varepsilon _1|,\ldots ,|\varepsilon _r|\}\), $$\begin{aligned} R_j({\mathbf {s}}):=\frac{s_j+\cdots +s_r-m_j-\cdots -m_r}{s_{j-1}+\cdots +s_r-m_{j-1}-\cdots -m_r} \end{aligned}$$, \(\varepsilon _1+\cdots +\varepsilon _r=0\), $$\begin{aligned} \zeta _r({\mathbf {s}})=\sum _{0\le h\le r-1}\sum _{2\le j_1 rapprox 1\), \(\zeta _t(s_1,\ldots ,s_{t-1},s_t+\cdots +s_r+k)\), \(\zeta (s_1+\cdots +s_r+k)=\zeta (m_1+\cdots +m_r+k)+O(\varepsilon )=(C_{\phi }+O(\varepsilon ))R_{\phi }({\mathbf {s}})\), $$\begin{aligned} \zeta (s_1+\cdots +s_r+k)&=\frac{1}{s_1+\cdots +s_r+k-1}+\gamma +O(\varepsilon )\\&=\frac{1+(\gamma +O(\varepsilon ))(\varepsilon _1+\cdots +\varepsilon _r)}{s_1+\cdots +s_r-m_1-\cdots -m_r}\\&=\frac{1}{s_1+\cdots +s_r-m_1-\cdots -m_r}(C_{\phi }+O(\varepsilon ))R_{\phi }({\mathbf {s}}) \end{aligned}$$, $$\begin{aligned}&\zeta (s_1+\cdots +s_r+k)\\&\quad =\zeta (m_1+\cdots +m_r+k)+O(\varepsilon )\\&\quad =\frac{1}{s_1+\cdots +s_r-m_1-\cdots -m_r}(\varepsilon _1+\cdots +\varepsilon _r) (\zeta (m_1+\cdots +m_r+k)+O(\varepsilon ))\\&\quad =\frac{1}{s_1+\cdots +s_r-m_1-\cdots -m_r}(0+O(\varepsilon ))R_{\phi }({\mathbf {s}}). 75 11 The Zeta Function of Riemann (Contd) 87 Teubner, Leibzig, 1929, Reprint: Chelsea (1965), Sasaki, Y.: Multiple zeta values for coordinatewise limits at non-positive integers. From specialfunctionswiki. Subscription will auto renew annually. To state the result of [11], we prepare some symbols and functions. Hereafter we prove it. We do not give the values of the constants \(C_{j_1j_2\cdots j_h}\) explicitly. Correspondence to \end{aligned}$$, $$\begin{aligned}&\zeta _t(s_1,\ldots ,s_{t-1},s_t+\cdots +s_r+k)\nonumber \\&\quad =\frac{1}{s_t+\cdots +s_r+k-1}\zeta _{t-1}(s_1,\ldots ,s_{t-2}, s_{t-1}+s_t+\cdots +s_r+k-1)\nonumber \\&\qquad +\sum _{k'=0}^{M-k-1}\left( {\begin{array}{c}-(s_t+\cdots +s_r+k)\\ k'\end{array}}\right) \zeta _{t-1}(s_1,\ldots ,s_{t-2},s_{t-1}+\cdots +s_r+k+k')\zeta (-k')\nonumber \\&\qquad +\frac{1}{\Gamma (s_t+\cdots +s_r+k)}I(s_1,\ldots ,s_{t-1}, s_t+\cdots +s_r+k;M-k-\eta )\nonumber \\&\quad =Z_1+Z_2+Z_3, \end{aligned}$$, \(M-k\ge M_t(m_1,\ldots ,m_{t-1},m_t+\cdots +m_r+k)+1\), $$\begin{aligned}&M_t(m_1,\ldots ,m_{t-1},m_t+\cdots +m_r+k)+1\\&\quad =\max _{1\le j\le t}\{t-j-m_j-\cdots -m_{t-1}-(m_t+\cdots +m_r+k)\}+1\\&\quad =\max _{1\le j\le t}\{r-j-m_j-\cdots -m_r\}+t-r-k+1\\&\quad =\max _{1\le j\le r}\{r-j-m_j-\cdots -m_r\}+t-r-k+1 \end{aligned}$$, $$\begin{aligned} =M_r({\mathbf {m}})+t-r-k+1\le M-k. \end{aligned}$$, $$\begin{aligned} Z_2&=\sum _{\begin{array}{c} 0\le k'\le M-k-1 \\ s_{t-1}+\cdots +s_r+k+k'\lessapprox 0 \end{array}} \left( {\begin{array}{c}-(s_t+\cdots +s_r+k)\\ k'\end{array}}\right) \nonumber \\&\quad \times \zeta _{t-1}(s_1,\ldots ,s_{t-2},s_{t-1} +\cdots +s_r+k+k')\zeta (-k')\nonumber \\&\quad +\sum _{\begin{array}{c} 0\le k'\le M-k-1 \\ s_{t-1}+\cdots +s_r+k+k' > rapprox 1 \end{array}} \left( {\begin{array}{c}-(s_t+\cdots +s_r+k)\\ k'\end{array}}\right) \nonumber \\&\quad \times \zeta _{t-1}(s_1,\ldots ,s_{t-2},s_{t-1}+\cdots +s_r+k+k') \zeta (-k')\nonumber \\&=Z_{21}+Z_{22}, \end{aligned}$$, \((s_{t-1}+\cdots +s_r-m_{t-1}-\cdots -m_r)^{-1}\), $$\begin{aligned}&\displaystyle \left( {\begin{array}{c}-(s_t+\cdots +s_r+k)\\ k'\end{array}}\right) \\&\quad \displaystyle =\frac{(-(s_t+\cdots +s_r+k)) \cdots (-(s_t+\cdots +s_r+k)-k'+1)}{k'!} Math. See the history of this page for a list of all contributions to it. A Laurent series for a meromorphic function f(z)f(z) at finite z∈ℂz\in\mathbb{C} has the form. replace Taylor series by Laurent series. When \(s_1+\cdots +s_r+k > rapprox 2\), we have. The Mellin–Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence. Bk({t}) with P1(t)=B1({t})={t}− 1 2 where Bk(t)is the kth Bernoulli polynomial and {t} is the fractional part of the real number t. Of particular interest as a Laurent series is the formal Dirac distribution. Tax calculation will be finalised during checkout. J. 61, 437–496 (2010), Matsumoto, K.: On the analytic continuation of various multiple zeta-functions. Ramanujan J. We divide \(\zeta _t(s_1,\ldots ,s_{t-1},s_t+s_{t+1}+\cdots +s_r+k)\) into 2 types; \(s_1,\ldots ,s_{t-1},s_t+s_{t+1}+\cdots +s_r+k\lessapprox 0\). The proof of this result is an immediate consequence of Theorem 2.1 in Section 2, which constructs the Laurent series of the inverse zeta function … When \(m_t+\cdots +m_r+k\ge 2\) and \(s_{t-1}+s_t+\cdots +s_r+k-1 > rapprox 1\), \(Z_1\) is represented as, The denominator \(s_t+\cdots +s_r+k-1\) is \(\not \sim 0\) and the factor. 52 A. Salem: Laurent Expansion of q-Zeta Functions where the reminder term Rℓ is defined as Rℓ=(−1)ℓ+1 Z n α Pℓ(t)f(ℓ)(t)dt Here, Bk denotes the kth Bernoulli number,and Pk(t)is the kth periodic Bernoulli function defined by Pk(t)= 1 k! !� -O“�>�Q^gX��ʪ.J˓���H �/�ą���U�2�)�<=-b�X����b�"z��[�4EO(Do�.��9��{%������9yzQ~��|/<9@��ʻ���[|tV���M�Ygh����I�4%�_�I���������K6��4���o]}ԵA��We��upZ��>l8�η�<6罯��z�^���7���?�e2�2��ydF��q�Ri�b� aa���N"5����#�!� 睆z�Ҝ�Q^��/5! Or, in some contexts one wants to take k=−∞k = -\infty. Würzburg Conf., Shaker, Aachen, pp. In: R. Steuding and J. Steuding (eds.) 6 0 obj Here such a formal sum (with powers extending infinitely in both directions) is suggestive notation for an element belonging to the dual ∏ n∈ℤR⋅z n\prod_{n \in \mathbb{Z}} R \cdot z^n of a ring ⊕ n∈ℤR⋅z n\oplus_{n \in \mathbb{Z}} R \cdot z^n (see Laurent polynomials below). %PDF-1.4 98, 107–116 (2001), MathSciNet  Z. For \({\mathbf {m}}\in ({\mathbb {Z}}_{\le 0})^r\) and \(k\in {\mathbb {Z}}\), we take an integer M satisfying \(k\le M-1\) and \(M\ge M_r({\mathbf {m}})+1\). Article  Γ ⁡ (z): gamma function, γ: Euler’s constant, ζ ⁡ (s): Riemann zeta function, π: the ratio of the circumference of a circle to its diameter, e: base of natural logarithm, s: complex variable and ρ: zeros Jump to: navigation, search. When \(m_t+\cdots +m_r+k=1\), \(Z_1\) is, Since \(s_{t-1}+s_t+\cdots +s_r+k-1\lessapprox 0\), \(\zeta _{t-1}\) has the expression of (A), so \(Z_1\) can be written in the form (B) in this case. 128, 1275–1283 (2000), Graduate School of Mathematics Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan, Multiple Zeta Research Center, Kyushu University, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan, Seikei University, Musashino-shi, Tokyo, 180-8633, Japan, You can also search for this author in More generally, for a∈k *a \in k^\ast an invertible unit, the series δ(az)=∑ n∈ℤa nz z\delta(a z) = \sum_{n \in \mathbb{Z}} a^n z^z satisfies, Just as it doesn’t make sense in general to multiply distributions (at least not without heavy qualifications, as in the theory of Colombeau), we cannot make sense of expressions like δ(z) 2\delta(z)^2. 8. 317–325 (2009), Zagier, D.: Values of zeta functions and their applications. Akiyama, S., Egami, S., Tanigawa, Y.: Analytic continuation of multiple zeta-functions and their values at non-positive integers. For discussion of products of distributions, see. (Proof of (\(A_t\)).) and so \(Z_{22}\) has the expression of (A) by the assumption. \end{aligned}$$, \((s_{t}+\cdots +s_r-m_{t}-\cdots -m_r)\sim 0\), $$\begin{aligned} -(s_t+\cdots +s_r+k)-k'+1=s_{t-1}-(s_{t-1}+s_t +\cdots +s_r+k+k'-1)\lessapprox 0. The variables in \(Z_{22}\) are of type 2. Tomokazu Onozuka. It has the following property: Indeed, notice that δ(z)\delta(z) is the distribution p↦p(1)p \mapsto p(1) which evaluates a Laurent polynomial at the multiplicative identity z=1z=1. The term \(Z_{22}\) has the factor \(1/(s_{t-1}+\cdots +s_r-m_{t-1}-\cdots -m_r)\) since \(\zeta _{t-1}\) satisfies (\(B_{t-1}\)). Observe that Laurent series in the generality discussed here do not analogously form a ring: the obvious definition of the coefficients of the product h(z)=f(z)g(z)h(z) = f(z)g(z) of two Laurent series f(z),g(z)f(z), g(z). A Laurent series in one variable zz over a commutative unital ring kk is a doubly infinite series. Math. J. Acta Arith. We use induction on t. We first prove the case \(t=1\). But here we emphasize the point of view that Laurent series have a distribution-like character, with Laurent polynomials being considered a space of functions ℤ→k\mathbb{Z} \to k with compact (finite) support, via the evident inclusion.


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